* Cheney's run renews concerns about party divisions
* In bid to take back Senate, party can afford few missteps
* Tension between party's establishment, Tea Party movement
By John Whitesides
WASHINGTON, July 18 (Reuters) - Liz Cheney's decision to challenge Wyoming U.S. Senator Mike Enzi in a Republican primary next year sets up the type of divisive, intraparty fight that Republican leaders vowed to avoid after the 2012 elections.
The move this week by Cheney, daughter of former Vice President Dick Cheney, is unlikely to open the door for a Democrat to win the Senate seat in Wyoming, which is heavily Republican.
But it is rekindling Republicans' concerns that party feuds could lead to a reprise of the 2010 and 2012 elections, when bitter Republican primary fights in several states produced weakened, gaffe-prone nominees who went on to lose winnable races to Democrats - and thwart Republicans' hopes of winning a majority in the Senate.
In next year's elections, Republicans will need a net gain of six seats to take control of the 100-member Senate. Most analysts expect the party to gain some seats - possibly enough to reclaim a majority - but say it cannot afford the types of missteps it has made in the past two elections.
The fields of candidates are still shaping up, but already there are signs of brewing Senate primary fights among Republicans in several states, including Iowa, Georgia and Alaska.
"The Republican path to success in the Senate is pretty narrow. If they lose one or two seats because of a difficult primary, that's a huge problem - and it's possible," said Jennifer Duffy, a Senate analyst for the nonpartisan Cook Political Report.
"They can't afford a lot of unforced errors," she added.
Many Republicans still have painful memories of the last two elections, when contentious primaries produced Senate nominees such as Todd Akin in Missouri, Richard Mourdock in Indiana, Sharron Angle in Nevada and Christine O'Donnell in Delaware.
Each rode the momentum of the conservative Tea Party movement to the Republican nomination, then withered amid campaign-trail missteps. Akin and Mourdock stumbled in 2012 by making controversial comments about rape.
During the 2010 campaign, Angle warned that Islamic law was taking hold in some U.S. cities, while O'Donnell - who previously had acknowledged dabbling in witchcraft - made a commercial to try to assure voters she was not a witch.
In the aftermath of the Republican losses last November, party leaders pledged to cut back on internal squabbling and focus on producing disciplined general election candidates.
Republican strategist Karl Rove announced a new political action committee, the Conservative Victory Project, that would back candidates deemed by party leaders to be the most "electable."
That drew criticism from Tea Party and other conservative groups, whose challenge to party leaders' authority is at the root of much of the tension within the Republican Party.
'FISSURES IN THE PARTY'
Cheney's announcement seemed to reinforce the message that some Republicans were not on board with the party's unification drive.
She implicitly criticized Enzi's sometimes accommodating style, saying on Tuesday that "we can no longer afford simply to go along to get along" - an echo of past party clashes between ideologically driven activists and more moderate party pragmatists.
"I think it's going to open a lot of fissures in the party," former Republican National Committee Chairman Michael Steele said on MSNBC. "I think this is an insurgent move by Cheney."
Republicans have some clear advantages heading into the 2014 campaign, including a playing field in which Democrats must defend 20 Senate seats to 14 for Republicans.
Democrats also are defending seven seats in states won by Republican Mitt Romney in the 2012 presidential election - Arkansas, Louisiana, Alaska, South Dakota, North Carolina, Montana and West Virginia.
But the prospect of ugly Republican nominating battles has given Democrats hope. In conservative Georgia, where Republican Senator Saxby Chambliss is retiring, several Republicans including three members of Congress - Paul Broun, Phil Gingrey and Jack Kingston - already have said they will seek the seat.
In politically divided Iowa, a few little-known Republican contenders are lining up to replace retiring Democratic Senator Tom Harkin. Democrats have rallied around U.S. Representative Bruce Braley as their candidate.
In Alaska, where Republicans are targeting Democratic Senator Mark Begich, Tea Party favorite Joe Miller is facing Lieutenant Governor Mead Treadwell in a race that could be complicated by the potential entry of another Republican: former Alaska Governor and 2008 vice presidential nominee Sarah Palin.
Palin said this month she was "considering" entering the race. Miller defeated Senator Lisa Murkowski in the 2010 Republican primary, only to lose to her in the general election after she mounted a write-in campaign.
SEEKING 'DISCIPLINED' CONSERVATIVES
Other states where competitive Republican primaries could develop include South Dakota - where former Governor Mike Rounds is running for retiring Democrat Tim Johnson's seat - and West Virginia, where conservatives are exploring a challenge to Republican Representative Shelley Moore Capito's bid for the Senate seat held by retiring Democrat Jay Rockefeller.
Two Republican senators who had been considered vulnerable to primary challenges - Senate Republican leader Mitch McConnell of Kentucky and Lindsey Graham of South Carolina - do not have strong primary opponents, but could still get them, analysts say.
Both have drawn the ire of conservatives at times. As his party's leader in the Senate, McConnell has been accused of not pushing a strong enough conservative agenda, while Graham has been criticized for straying from the party line - most recently in backing a plan to overhaul the country's immigration system.
"There is definitely a lot of frustration in the (conservative) grassroots and a real hunger for new leadership in Washington and new blood and new energy in the Senate," said Matt Hoskins, executive director of the Senate Conservatives Fund. The "Super PAC," or political action committee, once was led by former Senator Jim DeMint of South Carolina, an early booster of the Tea Party movement.
Hoskins said the fund was exploring whether to back conservatives in several states, including potential challengers to McConnell and Graham.
"In Kentucky, Republicans would be better off with a different Republican candidate. Mitch McConnell is a very weak general election candidate," Hoskins said.
Polls indicate McConnell is vulnerable in Kentucky, but he has nearly $10 million in campaign funds for his race against Democratic challenger Alison Lundergan Grimes, the secretary of state. McConnell also has formed an alliance with fellow Kentucky Republican Senator Rand Paul, a Tea Party favorite.
Jonathan Collegio, a spokesman for Rove's PAC, said the party's losses in 2012 had reinforced the need for a more disciplined approach to elections.
That new caution was most evident this year in Iowa, where staunchly conservative Representative Steve King chose not to enter the Senate race after some Republicans expressed concern that his views were out of step with most voters in the state.
"There appears to be a broad recognition now that in order to win general elections, we need high-quality candidates who aren't just conservatives but are also good fundraisers and disciplined campaigners," Collegio said.
But Hoskins said his group was still looking for credible conservatives who could win general elections.
"We have primaries for a reason. Republican voters should have choices and there should be a healthy debate," Hoskins said. "Once you are elected, you aren't entitled to be there forever. It's not a lifetime entitlement." (Editing by David Lindsey and Peter Cooney)